One-dimensional Euclidean Space Research Articles

Boundedness of composition operators on higher order Besov spaces in one dimension

This paper aims to characterize boundedness of composition operators on Besov spaces $$B^s_{p,q}$$ of higher order derivatives $$s>1+1/p$$ on the one-dimensional Euclidean space. In contrast to the lower order case $$0<s<1$$ , there were a few results on the boundedness of composition operators for $$s>1$$ . We prove a relation between the composition operators and pointwise multipliers of Besov spaces, and effectively use the characterizations of the pointwise multipliers. As a result, we obtain necessary and sufficient conditions for the boundedness of composition operators for general p, q, and s such that $$1<p\le \infty $$ , $$0<q\le \infty $$ , and $$s>1+1/p$$ . In this paper, we treat, as a map that induces the composition operator, not only a homeomorphism on the real line but also a continuous map whose number of elements of inverse images at any one point is bounded above. We also show a similar characterization of the boundedness of composition operators on Sobolev spaces.

Is Time the Real Line?

This paper is devoted to discussing the topological structure of the arrow of time. In the literature, it is often accepted that its algebraic and topological structures are that of a one-dimensional Euclidean space \(\mathbb {E}^1\), although a critical review on the subject is not easy to be found. Hence, leveraging on an operational approach, we collect evidences to identify it structurally as a normed vector space \((\mathbb {Q}, \vert \cdot \vert )\), and take a leap of abstraction to complete it, up to isometries, to the real line. During the development of the paper, the space-time is recognized as a fibration, with the fibers being the sets of simultaneous events. The corresponding topology is also exposed: open sets naturally arise within our construction, showing that the classical space-time is non-Hausdorff. The transition from relativistic to classical regimes is explored too.

Large deviations for the empirical measure of the zig-zag process

The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable for Monte Carlo simulation of continuous probability distributions. An important question in assessing the efficiency of this method is how fast the empirical measure converges to the stationary distribution of the process. In this paper we provide a partial answer to this question by characterizing the large deviations of the empirical measure from the stationary distribution. Based on the Feng–Kurtz approach, we develop an abstract framework aimed at encompassing piecewise deterministic Markov processes in position-velocity space. We derive explicit conditions for the zig-zag process to allow the Donsker–Varadhan variational formulation of the rate function, both for a compact setting (the torus) and one-dimensional Euclidean space. Finally we derive an explicit expression for the Donsker–Varadhan functional for the case of a compact state space and use this form of the rate function to address a key question concerning the optimal choice of the switching rate of the zig-zag process.

Inversion problems for Fourier transforms of particle distributions

Collective coordinates in a many-particle system are complex Fourier components of the local particle density , and often provide useful physical insights. However, given collective coordinates, it is desirable to infer the particle coordinates via inverse transformations. In principle, a sufficiently large set of collective coordinates are equivalent to particle coordinates, but the nonlinear relation between collective and particle coordinates makes the inversion procedure highly nontrivial. Given a ‘target’ configuration in one-dimensional (1D) Euclidean space, we investigate the minimal set of its collective coordinates that can be uniquely inverted into particle coordinates. For this purpose, we treat a finite number M of the real and/or the imaginary parts of collective coordinates of the target configuration as constraints, and then reconstruct ‘solution’ configurations whose collective coordinates satisfy these constraints. Both theoretical and numerical investigations reveal that the number of numerically distinct solutions depends sensitively on the chosen collective-coordinate constraints and target configurations. From detailed analysis, we conclude that collective coordinates at the smallest wavevectors is the minimal set of constraints for unique inversion, where represents the ceiling function. This result provides useful groundwork to the inverse transform of collective coordinates in higher-dimensional systems.

Entire solutions originating from monotone fronts to the Allen–Cahn equation

In this paper, we study entire solutions of the Allen–Cahn equation in one-dimensional Euclidean space. This equation is a scalar reaction–diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions originating from three and four fronts is shown by constructing some suitable pairs of super–sub-solutions. Moreover, we show that there are no entire solutions originating from more than four fronts.

Front propagation in nonlinear parabolic equations

We study existence and stability of travelling waves for nonlinear convection–diffusion equations in the one-dimensional Euclidean space. The diffusion coefficient depends on the gradient in analogy with the p-Laplacian and may be degenerate. We also prove that the solution converges to ± 1 outside an interface which moves with constant velocity; our results include both generation and propagation of interface properties. In particular, unconditional stability is established with respect to initial data perturbations in L 1 ( R ) .

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T)=O(k⋅n 1/k )⋅w(MST(M)), and a spanning tree T′ with weight w(T′)=O(k)⋅w(MST(M)) and unweighted diameter O(k⋅n 1/k ). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently. We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic one-dimensional Euclidean space. Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem in Computational Geometry. In STOC’95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n)⋅w(MST(M)). In SODA’05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors. Finally, our upper bounds imply improved approximation algorithms for the minimum h -hop spanning tree and bounded diameter minimum spanning tree problems for metric spaces.

Lax-Phillips Scattering for Atomorphic Functions Based on the Eisenstein Transform

We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincare upper half-plane by the full modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all functions with finite energy-form for the non-Euclidean wave equation. The use of the Eisenstein transform along with some properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.

Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data

In previous studies we have stated that the well-known bootstrap techniques are a valuable tool in testing statistical hypotheses about the means of fuzzy random variables, when these variables are supposed to take on a finite number of different values and these values being fuzzy subsets of the one-dimensional Euclidean space. In this paper we show that the one-sample method of testing about the mean of a fuzzy random variable can be extended to general ones (more precisely, to those whose range is not necessarily finite and whose values are fuzzy subsets of finite-dimensional Euclidean space). This extension is immediately developed by combining some tools in the literature, namely, bootstrap techniques on Banach spaces, a metric between fuzzy sets based on the support function, and an embedding of the space of fuzzy random variables into a Banach space which is based on the support function.

Fractal quantum propagators

Quantum time-evolution propagators are derived for a particle on a fractal lattice. It is shown that the path of such a particle represents a self-affine fractal. A relationship is obtained between the fractal dimension of the path (as determined by three different operational definitions) and the fractal dimension of the lattice. From these relationships it is seen that a quantum-mechanical particle in a one-dimensional Euclidean space also has a path which is a self-affine fractal. A strong analogy exists between these observations and the difference between Brownian motion and fractional Brownian motion. The uncertainty principle for a particle on a fractal lattice is derived and the phenomena of persistence and antipersistence is related to the uncertainty in momentum. The fractal quantum propagators are then used to develop a quantum theory of fractal rate constants. The theory of Miller and co-workers [J. Chem. Phys. 79, 4889 (1983); 90, 904 (1989)] is used to derive expressions for a bimolecular rate constant. Using the position-flux cross correlation function, scaling laws for the time dependence of the ‘‘rate constant’’ is obtained. These results are discussed in relation to experimental and theoretical results for time-dependent rate constants in percolation clusters.

INVERSE DYNAMIC PROGRAMMING II

This paper treats a class of finite-stage dynamic programming (DP) problems whose state space is an interval in the one-dimensional Euclidean space R1. By introducing a notion of “reward space” we can define, in this class, an inverse problem (called the inverse DP) to a given DP (called the main DP), because of the monotonicity and continuity of the reward functions and state transformations. Roughly speaking the inverse DP to the main DP is defined by replacing “max”, “reward functions” and “state transformations” with “min”, “inverse (in some sense) to the state transformations” and “inverse (in some sense) to the reward functions”, respectively, by replacing “terminal reward function” with “inverse to itself”, and by alternating “reward spaces” with “state spaces”, but by letting “action spaces” remain as they are. According to the monotonicity of the reward functions and state transformations we clarify the relation between the main DP and the inverse DP separately for four cases. Our main result is stated as follows. The pair of optimal reward functions and optimal policy for the main DP characterizes, in an inverse sense, the pair of optimal reward functions and optimal policy for the inverse DP, and vice versa (Section 2). Further we show that our DP represents a mathematical programming problem which has the property “recursiveness with monotonicity” in [5] and conversely such a problem can also be represented by our DP (Section 3). The last section is devoted to the illustration of several examples.

Complex-dimensional Integral and Light-cone Singularities

The notion of a complex-dimensional integral is introduced in the complex w-dimensional Minkowski space. Its basic properties, such as Lorentz invariance, are investigated. Complex-dimensional invariant delta functions dn(x',mz), A^n(x w2), etc. are explicitly calculated in position space. It is proposed to define products of singular functions in the ordinary Minkowski space by analytically continuing the corresponding ^-dimensional ones to w=4. The light-cone singularities of [^Oc;w2)]2, A(x;mz) r;w2) and [4(1) (x; ra2)]2 are shown to be unambiguously determined in this way. Recently, in quantum field theory, much attention has been paid to complex-dimensional regularization [1]. The momentum-space Feynman integral is regularized by considering it in the complex ^-dimensional space formally. The extension of the dimension 4 to the complex dimension n is easily done in the Feynman-parametric representation of the Feynman integral. The purpose of my talk is to formulate the theory of complex-dimen sional integrals in the general framework and apply it to regularizing singular products in position space. Detailed accounts are presented in my papers [2,3]. The complex ^-dimensional Minkowski space Mn is a product of a one-dimensional Euclidean space R and a complex (n — Y) -dimensional space En~l such that the scalar product in M.n is defined by the difference between the product in R and the scalar product in En~l. Here En~l is an

INVERSE DYNAMIC PROGRAMMING

This paper treats a class of finite-stage dynamic programming (DP) problems whose state space is an interval in the one-dimensional Euclidean space R1. By introducing a notion of “reward space” we can define, in this class, an inverse problem (called the inverse DP) to a given DP (called the main DP), because of the monotonicity and continuity of the reward functions and state transformations. Roughly speaking the inverse DP to the main DP is defined by replacing “max”, “reward functions” and “state transformations” with “min”, “inverse (in some sense) to the state transformations” and “inverse (in some sense) to the reward functions”, respectively, by replacing “terminal reward function” with “inverse to itself”, and by alternating “reward spaces” with “state spaces”, but by letting “action spaces” remain as they are. According to the monotonicity of the reward functions and state transformations we clarify the relation between the main DP and the inverse DP separately for four cases. Our main result is stated as follows. The pair of optimal reward functions and optimal policy for the main DP characterizes, in an inverse sense, the pair of optimal reward functions and optimal policy for the inverse DP, and vice versa (Section 2). Further we show that our DP represents a mathematical programming problem which has the property “recursiveness with monotonicity” in [5] and conversely such a problem can also be represented by our DP (Section 3). The last section is devoted to the illustration of several examples.

Sensitive Discount Optimality in Controlled One-Dimensional Diffusions

In this paper we consider the problem of optimally controlling a diffusion process on a compact interval in one-dimensional Euclidean Space. Under the assumptions that the action space is finite and the cost rate, drift and diffusion coefficients are piecewise analytic, we present a constructive proof that there exist piecewise constant $n$-discount optimal controls for all finite $n \geqq 1$ and measurable $\infty$-discount optimal controls. In addition we present a sequence of second order differential equations that characterize the coefficients of the Laurent series of the expected discounted cost of an $n$-discount optimal control.

Monotonicity of solutions of Volterra integral equations in Banach space

for almost all t. In this definition it is assumed, of course, that the integral on the right-hand side of (1.2) is in the domain of A. In a recent paper [6], Friedman and Shinbrot have studied the equation (1.1) even in the more general case where xo and A depend on t and -, respectively. They proved theorems of existence, uniqueness, differentiability and asymptotic behavior of solutions. They also constructed fundamental solutions and derived asymptotic bounds for them. We recall [6] that if xo is in the domain of A4, for some ,u>0, then the solution of (1.1) is given by S(t)xo. The purpose of the present paper is to derive monotonicity theorems for solutions of (1.1). We shall generalize some of the monotonicity theorems of Friedman [2] (see also [4]) from the case X= R1 (R1 the one-dimensional Euclidean space) to the case where X is any Banach space. In ?2 we give some auxiliary results. These results are concerned with Volterra equations in one-dimension (i.e., X=R1). In particular, we study the behavior of the solutions with respect to a certain parameter. In ?3 we give an integral formula for S(t) in case A is a bounded operator. For A unbounded, we construct a fundamental solution as a limit of fundamental solutions corresponding to the bounded operators A(I+ A/n) -1. We prove that the fundamental solution coincides with the fundamental solution of [6, Chapter 1]

Einstein Spaces Admitting a One-Parameter Group of Conformal Transformations

Publisher Summary This chapter considers M to be connected complete Einstein space of dimension n > 2 and of class C ∞ and suppose that a vector field on M generates globally a one parameter group of non-homothetic conformal transformations. Then M is isometric to a simply connected space of positive constant curvature. In particular M is homeomorphic to the sphere S n . If M is an irreducible and complete Riemannian manifold of class C ∞ , then A(M) is equal to I(M), except the case M is the one-dimensional euclidean space, where A(M) (I (M)) is the group of a fine (isometric) transformations of M. If a simply connected complete Riemannian manifold of class C ∞ admits a one-parameter group of non-isometric homothetic transformations, then it is a euclidean space.

On Poisson and composed Poisson stochastic set functions

Several investigations has recently been made concerning Poisson and composed Poisson stochastic processes. The ordinary Poisson process is conceivable as a sequence of points, distributed at random on the time axis and this idea can be generalized to more than onedimensional spaces. The latter case occurs in making a blood-count, in counting stars, etc. In [8], [4], [6] and [15] conditions are given ensuring the Poisson character of the distribution of the number of points in a set A of the one, resp. at least one-dimensional Euclidean space. In [8], [14], [1] and [13] similar problems are considered for the onedimensional Euclidean space and the main purpose is to prove that under some conditions the random variables ξt2 − ξt1 (t1 < t2) have composed Poisson distributions. We say that a random variable ξ has a composed Poisson distribution if its characteristic function f(u) can be written in the form

Uniformly best constant risk and minimax point estimates

Let#!, . . ., xn deno te s (not necessarily independent) observed values of a random variable f, which is distributed over a space S according to a distribution Pz(x, 0i, . . ., 0S). I t is assumed that Pz(x, 0i, . . ., 0S) is completely specified except for the s unknown parameters 0U . . ., 0S. These parameters may be represented by a point 6=(d1, . . ., 0S) in the s-dimensional Euclidean parameter space 12. Also X—(xi, . . ., xn) is a point in the n-dimensional Euclidean sample space, M. We shall assume that Pz(x, 0) is absolutely continuous, that is, £ possesses an integrable probability density function g^x, 0). Let p(X, 6)=p(x1, . . ., xn, 0) denote the joint probability density function of the observations at XeM. A statistical point estimate of a parameter 0*. which ranges over a subset co* of one-dimensional Euclidean space, is a function ft{X) of the sample values that takes on values in ut. Let W\fi(X), 0] be a nonnegative measurable function denned for all 0e& and XeM. W[fi(X), .0] is a weight function that represents the relative seriousness of taking ft(X) as the value of 0* for any particular sample point X. The function

On Unicoherency About a Simple Closed Curve

1. The subject matter of this article is a discussion of a property of point-sets which is closely related to the properties of being connected and of being a unicoherent continuum, and is in a sense a generalization of both these properties. A unicoherent continuum is defined by Kuratowski t as one which cannot be expressed as the union of two continua whose divisor is not connected, and this definition is in common usage. The analogy of the definition to that of a continuum is apparent. Ilowever, a set that is not connected may be connected between some pair of its points. We therefore propose a corresponding definition of unicoherency. A set 11 in a imetric space is unicoherent about the simple closed curve J if, for every decomposition of J into closed arcs h and k by points a and b and for every decomposition of M into relatively closed sets H and K such that h C iH, k C K., and h K = IH = a + b, ther-e is a component of H K containin.g both a and b. (If M is closed or is the space itself, the word relatively is to be omitted.) It is this extension of the idea of unicoherent continuum that is to be discussed. That this may be regarded also as a natural extension of the definition of connectivity between two points is apparent when we recall that in a one-dimensional Euclidean space an interval is a sphere and two points form its frontier or the surface of the sphere. t The property of a set being unicoherent about a simple closed curve is also useful in formulating an intrinsic definition of a two-dimensional simplex, as will be seen later. Readers of II. Whitney's recent article on this subject ? will note the marked resemblance to the property of a simple closed curve being homologous to zero in a closed set, which is the basis of his article. In fact, the present article grew out of a search for purely point-set properties equivalent to Whitney's definition.